Problem: Given $ m \angle RPS = 2x - 2$, $ m \angle QPR = 5x + 1$, and $ m \angle QPS = 97$, find $m\angle QPR$. $P$ $Q$ $S$ $R$
Answer: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {5x + 1} + {2x - 2} = {97}$ Combine like terms: $ 7x - 1 = 97$ Add $1$ to both sides: $ 7x = 98$ Divide both sides by $7$ to find $x$ $ x = 14$ Substitute $14$ for $x$ in the expression that was given for $m\angle QPR$ $ m\angle QPR = 5({14}) + 1$ Simplify: $ {m\angle QPR = 70 + 1}$ So ${m\angle QPR = 71}$.